Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview

Abstract

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha \in (0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.

arXiv comment24 pages, 3 figures

References in corpus (15)

• A new difference scheme for the time fractional diffusion equation
• Ten equivalent definitions of the fractional Laplace operator
• On an explicit finite difference method for fractional diffusion equations
• A Tutorial on Inverse Problems for Anomalous Diffusion Processes
• Stochastic modeling in nanoscale biophysics: Subdiffusion within proteins
• WSLD operators: A class of fourth order difference approximations for space Riemann-Liouville derivative
• A Parabolic Problem with a Fractional-Time Derivative
• Numerical Methods for Fractional Diffusion
• Strong maximum principle for fractional diffusion equations and an application to an inverse source problem
• Too much regularity may force too much uniqueness
• Time-stepping error bounds for fractional diffusion problems with non-smooth initial data
• Numerical Approximations for Fractional Differential Equations
• Numerical solution of parabolic problems based on a weak space-time formulation
• An $L_q(L_p)$-theory for the time fractional evolution equations with variable coefficients
• Finite element approximation of a time-fractional diffusion problem in a non-convex polygonal domain

Cited by in corpus (33)

• Error analysis for a fractional-derivative parabolic problem on quasi-graded meshes using barrier functions
• A preconditioning technique for an all-at-once system from Volterra subdiffusion equations with graded time steps
• A fast implicit difference scheme for solving the generalized time-space fractional diffusion equations with variable coefficients
• A parallel-in-time two-sided preconditioning for all-at-once system from a non-local evolutionary equation with weakly singular kernel
• Two time-stepping schemes for sub-diffusion equations with singular source terms
• A linear Galerkin numerical method for a quasilinear subdiffusion equation
• Fractional time stepping and adjoint based gradient computation in an inverse problem for a fractionally damped wave equation
• Initial-boundary value problems to semilinear multi-term fractional differential equations
• Identifying source term in the subdiffusion equation with L^2-TV regularization
• Optimal error analysis of a non-uniform IMEX-L1 finite element method for time fractional PDEs and PIDEs
• Analysis of a WSGD scheme for backward fractional Feynman-Kac equation with nonsmooth data
• Sharp pointwise-in-time error estimate of L1 scheme for nonlinear subdiffusion equations
• Homogenization of time-fractional diffusion equations with periodic coefficients
• The Best Uniform Rational Approximation: Applications to Solving Equations Involving Fractional powers of Elliptic Operators
• Total variation regularization for recovering the spatial source term in a time-fractional diffusion equation
• Multi-term fractional linear equations modeling oxygen subdiffusion through capillaries
• Numerical scheme for the Fokker-Planck equations describing anomalous diffusions with two internal states
• Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type
• Approximation of an optimal control problem for the time-fractional Fokker-Planck equation
• Complete monotonicity-preserving numerical methods for time fractional ODEs
• Wasserstein Gradient Flow Formulation of the Time-Fractional Fokker-Planck Equation
• Incomplete Iterative Solution of the Subdiffusion Problem
• Strong convergence order for the scheme of fractional diffusion equation driven by fractional Gaussion noise
• High-order Time Stepping Schemes for Semilinear Subdiffusion Equations
• Subdiffusion with a time-dependent coefficient: analysis and numerical solution
• Numerical Estimation of a Diffusion Coefficient in Subdiffusion
• Reconstruction of a Space-Time Dependent Source in Subdiffusion Models via a Perturbation Approach
• Subdiffusion with Time-Dependent Coefficients: Improved Regularity and Second-Order Time Stepping
• Uniform estimates for a fully discrete scheme integrating the linear heat equation on a bounded interval with pure Neumann boundary conditions
• On a Non-Uniform $\alpha$-Robust IMEX-L1 Mixed FEM for Time-Fractional PIDEs
• Abstract Fractional Cauchy Problem: Existence of Propagators and Inhomogeneous Solution Representation
• Galerkin Type Methods for Semilinear Time-Fractional Diffusion Problems
• An Inverse Potential Problem for Subdiffusion: Stability and Reconstruction